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This work describes a transmittance model that evaluates global solar irradiation through the atmospheric column and at surface. The model is based on appropriate determination of the transmission coefficients of the different atmospheric constituents in a plane parallel layers frame and estimates the downward solar fluxes from the upper limit of the atmosphere. In testing this model, we first considered the purely molecular atmosphere to parameterize descending solar fluxes, which allowed us to estimate the attenuation due to atmospheric gases at specific times of the day when the irradiation at ground level is known. The results thus obtained show that the molecular atmosphere has a maximum reduction rate of incident flux (at the Top Of Atmosphere) of 20% with daily profiles that are homogeneous with extraterrestrial fluxes. Considering the turbid and cloudy atmosphere in which the multiple scattering phenomenon is taken into account, we obtain at ground level fluctuating profiles with attenuation rates reaching 64% depending on the time instant in the day. The comparison of our results with the experimental data obtained at the Yaoundé site on one hand and with the results of the CLIRAD-SW model on the other hand shows at monthly scale high correlation, of the order of 0.998. Moreover at monthly time scale, the precision which for some hourly values is relatively low tends towards a net improvement on the seasonal scale where it extends over a narrow domain ranging from 0.02% to 1.66%.

Thanks to the radiative impact of the atmospheric constituents, the solar radiation is attenuated as it travels through the terrestrial atmosphere. This attenuation is due to absorption by gases, scattering and absorption by solid (aerosols) and liquid (clouds) particles. During the last decades, numerous studies aiming at better understanding the nature of the atmospheric constituents, their origin as well as their radiative impact for appropriate consideration in the determination of energy transfer through the atmosphere were carried out [^{st} century. Other researchers have focused on the determination of microphysical and optical properties through in situ measurements in order to obtain a more precise evaluation of these properties [

Nevertheless, these methods and models may still exhibit some shortcomings in the sense that satellites have a limited time coverage and their estimations of surface shortwave irradiation does not agree with ground-based measurements in all localities [

In this work, we propose to follow the downward solar radiation in different atmospheric layers located at various pressure levels from the Top of Atmosphere (TOA) to the surface, in order to determine the solar fluxes at altitude and surface in a plane parallel tropical atmosphere model that takes into account atmospheric gases, clouds and aerosols. The model resolves the diurnal cycle and cloud and aerosol fields are updated hourly [

The atmosphere consists of several radiatively active gases (N_{2}, O_{2}, O_{3}, H_{2}O, CO_{2}...) whose radiative properties are based typically on the determination of the absorption coefficients. These coefficients in turn depend on the intensity of the spectral lines and their broadening profiles. The main factors affecting the line spectrum are pressure, temperature, and molecular density, molar fraction of an active species and that of other species that act as upsettings in intermolecular collisions. Combining Boltzmann’s law on energy levels distribution, Planck’s law on monochromatic radiation and the Einstein’s theory of molecular absorption, we obtain the expression

S η = 8 π 3 η 3 h g l P k T Q ( T ) P 0 2 ( 1 − e − h C 0 η k T ) e − E l k T (1)

where S η is the intensity of the spectral line, η the wave number (cm^{−1}), h the Planck constant, k the Boltzmann constant, T the temperature, P the pressure. Q(T) is the partition function of the molecule, P 0 the dipolar moment of the molecule, g l the degeneracy of the level l, C 0 the celerity of the light in the vacuum and E l the energy of the level l.

To determine the intensity of spectral lines in a ro-vibrational band, we consider the approximations of harmonic oscillator and rigid rotator, with additional assumptions that the bandwidth is small compared to the wave number at the center of the band and that only the P and R branches are important [

S η P = 8 π 3 η P 3 h g l P k T Q ( T ) P 0 2 ( 1 − e − h C 0 η P k T ) e − E l k T (2)

With η P = η 0 − ( B v + 1 + B v ) j + ( B v + 1 − B v ) j 2 , j = 1 , 2 , 3 , ⋯

S η R = 8 π 3 η R 3 h g l P k T Q ( T ) P 0 2 ( 1 − e − h C 0 η R k T ) e − E l k T (3)

With

η R = η 0 + 2 B v + 1 + ( 3 B v + 1 + B v ) j + ( B v + 1 − B v ) j 2 , j = 1 , 2 , 3 ,

where B v is the moment of inertia of the molecule that depends on the vibrational energy level.

The profiles described above for the P and R bands consider that the spectral line is constant and centered on η. This does not reflect the reality because due to thermophysical conditions of the medium the line undergoes broadening phenomenon whose profiles have been described by Lorentz and Doppler among others. The broadening profile that we consider here is Voigt’s profile because it tends to narrower profiles at low pressures and to wider profiles at high pressures in harmony with Lorentz and Doppler. The empirical Voigt profile, often used for trace gas detection [

ϕ V ( η − η 0 ) = ϕ V ( η 0 ) ( ( 1 − x ) exp ( − 0.693 y 2 ) + x 1 + y 2 + β ) (4)

where

β = 0.016 ( 1 − x ) x ( exp ( − 0.0841 y 2.25 ) − 1 1 + 0.0210 y 2.25 ) (5)

with x = γ L γ V and y = | η − η 0 | γ V .

γ L and γ D are respectively the half-widths of Lorentz and Doppler profiles. γ L increases with pressure and decreases with temperature while γ D increases with temperature.

ϕ V ( η 0 ) = 1 2 γ V ( 1.065 + 0.447 x + 0.058 x 2 ) (6)

The half-width γ V of the Voigt profile is given by

γ V = 0.5346 γ L + ( 0.2166 γ L 2 + γ D 2 ) 1 / 2 (7)

Following this expression, the Voigt profile at high-pressure tends to Lorentz profile, when x tends to 1 and to Gaussian profile at low-pressure, when x tends to 0.

Typical profile half widths for atmospheric species at various pressure levels follow Voigt’s profile which is in the range 5 × 10^{−3} to 2 × 10^{−2} cm^{−1}, quite less than the distance between rotational transitions of many molecules of atmospheric interest. Thus the use of Voigt broadening makes it possible to increase the sensitivity and the selectivity of the line to be absorbed. It then becomes possible to make measurements in the spectral regions where H_{2}O and CO_{2} have strong and weak absorptions. The absorption coefficient of a molecule can then be written as

k η = S η ⋅ ϕ V ( η − η 0 ) (8)

where ϕ V ( η − η 0 ) is the Voigt broadening profile.

It follows that the transmittance of a gas layer can be expressed as

τ = exp ( − k η N Δ P ρ g ) (9)

where N is the number of gas molecules in the layer, ΔP (mbar) the pressure variation across the layer, ρ (kg∙m^{−3}) the density of the gas, g (N/kg) the specific gravity and k η (cm^{−1}) the average Absorption coefficient of the molecules.

Mc Cartney [

σ R λ = 24 π 3 N 0 2 λ − 4 ( n 0 2 − 1 n 0 2 + 2 ) 2 ( 6 + 3 ξ 6 − 7 ξ ) (10)

where ξ is the depolarization factor, N 0 the density of molecules (2.547305 × 10^{25} m^{−3} at 15˚C), n 0 the integer part of the refractive index of air and λ the wavelength. The cross section σ R λ is typically given in units of square centimeters.

For a given volume of gas, the total Rayleigh scattering coefficient φ is given by the product of the Rayleigh cross section per molecule and the molecular density N at a given level of altitude Z, pressure P and temperature T [

φ ( z , λ ) = N ( z ) σ R λ ( λ ) (11)

The Rayleigh optical thickness at altitude Z 0 is then given as the integral of the total volume scattering coefficient φ from Z 0 to the TOA, according to the relationship

τ R λ ( λ ) = ∫ Z 0 ∞ φ ( z , λ ) d z (12)

This equation was reevaluated using the most recent determinations of ξ = 0.0279 [

τ R λ = P / ( a 1 λ 4 + a 2 λ 2 + a 3 + a 4 λ − 2 ) (13)

where P is the relative pressure defined by P = p / p 0 with p being the local pressure, p_{0} the atmospheric pressure at surface, a 1 = 117.2594 μ m − 4 , a 2 = − 1.3215 μ m − 2 , a 3 = 3.2073 × 10 − 4 , a 4 = − 7.6842 × 10 − 5 μ m 2

Equation (13) reproduces the values obtained with Equation (12) with a difference of less than 0.01% over the entire spectrum [

The impact of clouds (greenhouse effect, albedo…) on the atmospheric radiative budget apart from their spatial and temporal coverage depends on their microphysical (Effective radius, Liquid and Ice Water Contents) and optical (optical thickness, albedo, asymmetry factor) properties [

k e x ( λ , m ) = ∫ 0 ∞ π r 2 S e x ( r , λ , m ) n ( r ) d r (14)

where S e x denotes the extinction efficiency which quantifies the attenuation of the incident radiation by a particle, λ the wavelength, m the complex refractive index, n(r) the particle density and r the radius of the particle. This extinction is related to the liquid water content (LWC) by the relationship

LWC k e x = 4 3 ρ w ∫ 0 ∞ π r 3 n ( r ) d r ∫ 0 ∞ π r 2 S e x ( r , λ , m ) n ( r ) d r (15)

In the case of a cloud consisting of large water droplets (spherical shape), S e x tends to 2 in the visible domain [

LWC k e x = 4 3 ρ w ∫ 0 ∞ π r 3 n ( r ) d r 2 ∫ 0 ∞ π r 2 n ( r ) d r (16)

But the effective radius of a cloud particle is defined as r e f f = ∫ 0 ∞ r 3 n ( r ) d r ∫ 0 ∞ r 2 n ( r ) d r then,

k e x = 3 LWC 2 ρ w r e f f (17)

It follows that the extinction coefficient is proportional to the ratio of the water content to the effective radius of the particle. This coefficient is then integrated over the thickness of the atmospheric column to deduce the optical thickness τ which quantifies the attenuation of the radiation along a path in the cloud. It reflects the ability of the medium to absorb and scatter this radiation.

τ = ∫ 0 h σ e x ( λ , m , Z ) d Z (18)

This work integrates three types of clouds as described by Akana and Njomo [

To produce a realistic parameterization of the irradiation, a certain number of approximations are made as regards the vertical variability of the density of atmospheric particles and the spectral variability of their optical properties. In this study, the heterogeneous atmosphere is discretized in 50 plane and parallel layers at different pressure levels. This structure corresponds to the AFGL profile [

For each cloudy and turbid atmospheric layer and for each spectral band, the effective optical thickness, single scattering albedo and asymmetry factor are calculated. The transmissivity T and the reflectivity R of a layer illuminated by a direct beam are calculated from the δ-Eddington approximation. This operation uses the discrete ordinate algorithm of Stamnes et al. [

pressure level is 1 2 [ P i + P i + 1 ] using two-stream method. By separating the direct

and diffuse components of the radiation, the expression of the transmittance of the layer is given by [

T a b ( μ 0 ) = e − τ a / μ 0 T b ( μ 0 ) + T ¯ b { e − τ a μ 0 R ¯ a R b ( μ 0 ) + [ T a b ( μ 0 ) − e − τ a μ 0 ] } / ( 1 − R ¯ a R ¯ b ) (19)

where T a b ( μ 0 ) , e − τ a / μ 0 and T a b ( μ 0 ) − e − τ a μ 0 are the total, direct and diffuse

transmittances respectively. R a and R b are direct reflectances of the sub-layers (a) and (b) respectively, R ¯ a and R ¯ b are the reflectances of these sub-layers respectively when they are illuminated by diffuse radiation.

The solar flux that emerges from an atmospheric layer is expressed as I L = τ I 0 where τ is the product of the transmitivities of all species contained in the layer. For a layer containing n species, we thus obtain τ = ∏ i = 1 n τ i where τ i is the transmittance of species i in a given spectral band. The solar spectrum which extends from 0.175 μm to about 10 μm is divided into spectral bands according to the model described above to better integrate the spectral behavior of atmospheric molecules in the attenuation of solar fluxes. If Γ is the total transmittance of the layer for all n gases and in all m spectral bands, then the overall flux transmitted is I 1 = Γ ⋅ I 0 with

Γ = ∑ j = 1 m r j ( ∏ i = 1 n τ i ) (20)

where r j is the fraction of extraterrestrial solar flux in the band j. The transmitted flux is therefore

I 1 = I 0 ∑ j = 1 m r j ( ∏ i = 1 n τ i ) (21)

This emerging flux is taken as incident to the adjacent layer. Thus, the solar flux emerging at the k^{th} layer is

I k = I 0 ⋅ ∏ k = 1 n Γ k (22)

The downward flux is thus calculated step by step through the 50 layers under different atmospheric conditions and then compared to the ground-based data measured by Laboratoire de Recherche Energétique (LRE, Yaoundé) [

Measurements of total and diffuse solar radiation have been conducted in Cameroon using Eppley PSP Pyranometers distributed in ten meteorological stations across the country. The accuracy of these devices calibrated for hemispherical integration was ±3% - 4%. The devices were producing incident solar flux with 10 seconds resolution. Data were then averaged over a period one hour and stored as the value of the total or diffuse flux for the corresponding hour [

The daily variations of simulated and measured global solar fluxes are presented for each month of the experimentation period. The measured data are the monthly hourly averages of the global radiation [

of the day, 20% at best. The very significant difference observed between simulations and measurements during these hours and over the four months reveals the importance of clouds and aerosols in the modulation of fluxes through the atmosphere. Indeed, in the absence of these, radiation modulation through the atmospheric column is globally poor and the fluxes received at surface are simply the expression of a limit or extreme situation which is very often used as standard in the prediction of solar fluxes. This atmospheric model is therefore not suitable for discussing the accuracy of radiative models.

Following the limitation of molecular atmosphere as indicated previously, we first studied separately the impact of aerosols and clouds in the attenuation of solar irradiation for the same periods.

There is a weaker reduction of around 36% in the middle of the day and for every month that is naturally due to the intense sunshine at that period. However, the maximum cloud reduction noted during April (rainy season) can be the result of high cloud density that quantitatively highlights optical thicknesses and backward scattering.

Our model is then used to infer solar fluxes in the case of a real atmosphere. The detailed calculations of the simulations extend over the spectral range [0.175 μm - 10 μm] and integrate all atmospheric components radiatively active in this spectrum (gas, clouds and aerosols).

Furthermore, we illustrate in

The main features of this regression analysis are summarized in

This paragraph, with regard to the indeed small differences between our simulations and the measurements on a monthly time scale, is concerned with seasonal distributions of fluxes at surface. In fact, seasonal, annual or interannual trends

Month | Correlation Coefficient | Relative error en % | Regression Coefficient |
---|---|---|---|

January | 0.9837 | 1.63 | 0.8907 |

February | 0.9882 | 1.18 | 0.9676 |

March | 0.9945 | 0.55 | 0.9533 |

April | 0.9945 | 0.55 | 0.9862 |

May | 0.9899 | 1.01 | 0.8987 |

June | 0.988 | 1.2 | 0.9012 |

July | 0.9835 | 1.65 | 0.9447 |

August | 0.9721 | 2.79 | 0.9169 |

September | 0.9881 | 1.19 | 1.0095 |

October | 0.9866 | 1.34 | 0.8401 |

November | 0.9529 | 4.72 | 0.8604 |

December | 0.9827 | 1.73 | 0.8348 |

in climate study most often contain more realistic information than simple monthly variations. Note in this context that Yaoundé has four seasons including the little rainy season (March-June), the heavy rainy season (September-November), the little dry season (July-August) and the heavy dry season (December to February).

One can notice from the table that the relative error is more comfortable at the seasonal time scale with a maximum of 1.66%, indicating a clear improvement over the monthly distribution. Similarly, correlation and regression coefficients increase significantly and this can be seen as a sign of a good accuracy of our simulations. This trend is confirmed by

Seasons | Correlation Coefficient | Relative error (%) | Regression Coefficient |
---|---|---|---|

Little rainy Season | 0.9998 | 0.02 | 0.9082 |

Little dry Season | 0.9834 | 1.66 | 0.9369 |

Heavy Rainy Season | 0.9897 | 1.03 | 0.9201 |

Heavy Dry Season | 0.9937 | 0.63 | 0.9069 |

However, it is noticeable that for the little dry season, although the correlation is good, the flux difference that existed at the monthly scale persists for the season. This would be due to the presence of a heavy cloud cover during the month of August that is generally observed on the site and that covers almost all the southern part of Cameroon due to the Intertropical Convergence Zone (ICZ). For the rest of seasons, a slight difference persists between the data in the time slot from 11:00 am to 13:00 pm. It should be noted that contrary to the monthly distributions, the time intervals where this discrepancy is noted become confined as we move to seasonal profiles. Finally, the monthly as well as the seasonal trends in harmony with the measurements and the computation time of our model which is shorter compared to CLIRAD-SW could viewed as key factors indicating that this model is suitable for the evaluation of the vertical profiles of solar irradiation through the atmosphere. However, further work is needed on spectral integration to refine the optical characterization of water vapor in the infrared region.

This study is a step towards the construction of a simple and flexible parameterized model for the evaluation of short wavelength fluxes through the atmosphere. It aims at regionally documenting the daily and seasonal variations of the solar fluxes necessary not only for an adequate dimensioning of the solar systems of energy generation but also for a more accurate characterization of the local variations of the atmospheric temperature, relevant factor of the equilibrium of the Earth’s climate. Atmospheric gases are described by their spectral absorption coefficients whereas clouds and aerosols are by their optical thicknesses and simple scattering albedo. Unlike other models, the spectral properties of the gases are directly calculated in the code. This significantly reduces the calculation time. Since our atmosphere is discretized vertically in layers, the two-flux method allows us to determine the transmittance of each of them in order to evaluate fluxes. The results generated by this model for the city of Yaoundé correlate well with measurements made at ground level on a monthly time scale and more on a seasonal scale (correlation coefficients > 0.99 ). Similarly, the relative differences between the simulation results and the measurements are on the average of 0.55% monthly and 0.02% seasonally. Despite these indicators of the quality of the model, it would be advisable to improve it especially with regard to the water vapor in the infrared in order to obtain a precision comparable to rigorous models.

The authors declare no conflicts of interest regarding the publication of this paper.

Tchouankap, J.D. and Nguimdo, L.A. (2020) Parameterized Transmittance Model for Atmospheric and Surface Solar Radiations. Atmospheric and Climate Sciences, 10, 81-99. https://doi.org/10.4236/acs.2020.101004